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In Mathematics, the adjective "linear" is used in two main senses:

  1. A function φ on a vector space V over a field F is called linear iff:
    • ∀x,y∈V: φ(x+y)=φ(x)+φ(y)
    • ∀x∈V,a∈F: φ(a*x)=a*φ(x).
    For a function over the field R, the second condition will follow if φ is also known to be continuous.

I've intentionally said nothing about the range of φ. When the range is F, φ is a linear functional; when it is another vector space W, φ is a linear transformation. The adjective might be applied in other situations (even when not working on a vector space!), too. For instance, a little-known fact is that perimeter is linear for convex shapes in R2!

  • Confusingly, a function l:R->Rk is also called "linear" if it describes a straight line: l(x)=x*a+b.
  • When b!=0, l is NOT "linear" in the first sense! It would more properly be called an affine function. Unfortunately we're stuck with the older name.